\section{Implementation}
Avoid matrix multiplications. Eq.~\ref{eq:inverseStateSpaceInput}.
\begin{align*}
	\vec{F} &= \dfrac{1}{4}\begin{pmatrix}
	1 &  0 & -2 &  1 \\
	1 &  2 &  0 & -1 \\
	1 &  0 &  2 &  1 \\
	1 & -2 &  0 & -1
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
	1 & 0 & 0 & 0 \\
	0 & \dfrac{1}{l} & 0 & 0 \\
	0 & 0 & \dfrac{1}{l} & 0 \\
	0 & 0 & 0 & \dfrac{\text{d}F}{\text{d}M}
	\end{pmatrix} \cdot
	\begin{pmatrix}
	m & 0 \\
	0 & \Theta
	\end{pmatrix} \cdot \vec{u}
	\\
	&= \dfrac{1}{4}\begin{pmatrix}
	1 &  0 & -2 &  1 \\
	1 &  2 &  0 & -1 \\
	1 &  0 &  2 &  1 \\
	1 & -2 &  0 & -1
	\end{pmatrix} 
	\cdot 
	\begin{pmatrix}
	m & 0 & 0 & 0 \\
	0 & \dfrac{\theta_{11}}{l} & 0 & 0 \\
	0 & 0 & \dfrac{\theta_{22}}{l} & 0 \\
	0 & 0 & 0 & \dfrac{\text{d}F}{\text{d}M} \cdot \theta_{33}
	\end{pmatrix} \cdot \vec{u}
	\\
	&= \frac{1}{4} \cdot 
	\begin{pmatrix}
		m & 0 & -2\cdot \dfrac{\theta_{22}}{l} & \frac{\text{d}F}{\text{d}M} \cdot \theta_{33}
		\\
		m & 2\cdot \dfrac{\theta_{11}}{l} & 0 & -\frac{\text{d}F}{\text{d}M} \cdot \theta_{33}
		\\
		m & 0 & 2\cdot \dfrac{\theta_{22}}{l} & \frac{\text{d}F}{\text{d}M} \cdot \theta_{33}
		\\
		m & -2\cdot \dfrac{\theta_{11}}{l} & 0 & -\frac{\text{d}F}{\text{d}M} \cdot \theta_{33}
	\end{pmatrix} \cdot \vec{u}
	\\
	&= \frac{1}{4}\cdot \begin{pmatrix}
		m\cdot u_1 - 2\cdot \dfrac{\theta_{22}}{l} \cdot u_3 + \frac{\text{d}F}{\text{d}M} \cdot \theta_{33} \cdot u_4 
		\\
		m\cdot u_1 + 2\cdot \dfrac{\theta_{11}}{l} \cdot u_2 - \frac{\text{d}F}{\text{d}M} \cdot \theta_{33} \cdot u_4
		\\
		m\cdot u_1 + 2\cdot \dfrac{\theta_{22}}{l} \cdot u_3 + \frac{\text{d}F}{\text{d}M} \cdot \theta_{33} \cdot u_4
		\\
		m\cdot u_1 - 2\cdot \dfrac{\theta_{11}}{l} \cdot u_2 - \frac{\text{d}F}{\text{d}M} \cdot \theta_{33} \cdot u_4
	\end{pmatrix} 
\end{align*}

\clearpage